On the solution of stochastic multiobjective integer linear programming problems with a parametric study

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1 O the soluto of stochastc multobectve tege lea pogammg poblems wth a paametc study Oma M. Saad ad Osama E. Imam Depatmet of Mathematcs, Faculty of Scece, Helwa Uvesty, Egypt. Depatmet of Ifomato Systems, Faculty of Compute Scece, Helwa Uvesty, Egypt. Abstact I ths study we cosde a multobectve tege lea stochastc pogammg poblem wth dvdual chace costats. We assume that thee s adomess the ght-had sdes of the costats oly ad that the adom vaables ae omally dstbuted. Some stablty otos fo such poblem ae chaactezed. A aulay poblem s dscussed ad a algothm as well as a eample s peseted. Key Wods: Multobectve tege lea pogammg; chace-costaed techque; Bach-ad Boud method; Stablty.. INTRODUCTION Decso poblems of stochastc o pobablstc optmzato ase whe ceta coeffcet of a optmzato model ae ot fed o kow but ae stead, to some etet, stochastc(o adom o pobablstc) quattes. I ecet yeas methods of multobectve stochastc optmzato have become ceasgly mpotat scetfcally based decso-makg volved pactcal poblems asg ecoomc, dusty, health cae, taspotato, agcultue, mltay puposes ad techology. We efe the Stochastc pogammg Web Ste (00)[0] fo lks to softwae as well as test poblem collectos fo stochastc pogammg. I addto, we should pot the eade to a etesve lst of papes mataed by Maate va de Vlek at the Web Ste: mally.eco.ug.l /bblo/ SP lst.html. I lteatue thee ae may papes that deal wth stablty of solutos fo stochastc multobectve optmzato poblems. Amog the may suggested appoaches fo teatg stablty fo these poblems [, 4, 8, 5]. Coespodg autho E-mal: omasd55@hotmal.com

2 Moe ecetly, some papes fo the autho ad othes have bee publshed the aea of stochastc multobectve optmzato poblems, fo eample, [9,0,]. I [9], a soluto algothm s peseted fo solvg tege lea pogammg poblems volvg depedet adom paametes the obectve fuctos ad wth lealy depedet adom paametes the costats. The ma featue of the poposed algothm s based maly upo the chace-costaed pogammg techque [] alog wth the cuttg-plae method of Gomoy [4]. Saad [0] evewed theoy ad methodology that have bee developed to cope wth the complety of optmzato poblems ude ucetaty. The classcal ecouse-based stochastc pogammg, obust stochastc pogammg, pobablstc pogammg have bee dscussed ad cotasted. I addto, the advatages ad shotcomgs of these models ae evewed. Applcatos ad the state-of-the-at computatos ae also suveyed ad seveal ma aeas fo futue developmet ths feld ae epoted. Stablty of soluto multobectve tege lea pogammg poblems s vestgated [], whee the poblem volves adom paametes the ght-had sde of the costats oly ad those adom paametes ae omally dstbuted. Some stablty otos fo such poblems have bee also chaactezed. Ths pape s ogazed as follows: we stat Secto by fomulatg the model of chacecostaed multobectve tege lea pogammg poblem (CHMOILP) ad the soluto cocept s toduced. I Secto, a paametc study s caed out o the poblem of coce, whee some basc stablty otos ae chaactezed fo the fomulated model. These otos ae the set of feasble paametes; the solvablty set, ad the stablty set of the fst kd (SSK). Moeove, a algothm s descbed to deteme the (SSK) fo the (CHMOILP). I Secto 4, a eample s povded to llustate the developed esults. Fally, Secto 5, some ope pots ae stated fo futue eseach wok the aea of stochastc multobectve tege optmzato poblems.. PROBLEM STATEMENT AND THE SOLUTION CONCEPT The chace-costaed multobectve tege lea pogammg poblem wth adom paametes the ght-had sde of the costats ca be stated as follows: (CHMOILP): whee ma F(), subect to X,

3 X = R P{ g () a b } α, =,,..., m, 0 ad tege, =,,.. =. Hee s the vecto of tege decso vaables ad F() s a vecto of k-lea eal-valued obectve fuctos to be mamzed. Futhemoe, P meas pobablty ad α s a specfed pobablty value. Ths meas that the lea costats may be volated some of the tme ad at most 00(- α ) % of the tme. Fo the sake of smplcty, we assume that the adom paametes b, ( =,, m) ae dstbuted omally wth kow meas E{b } ad vaaces Va {b } ad depedetly of each othe. Defto. A pot X s sad to be a effcet soluto fo poblem (CHMOILP) f thee does ot est aothe X such that F() F( ) ad F() F( ) wth P { g ( ) a b } α,,,..., m. = = The basc dea teatg poblem (CHMOILP) s to covet the pobablstc atue of ths poblem to a detemstc fom. Hee, the dea of employg detemstc veso wll be llustated by usg the teestg techque of chace-costaed pogammg []. I ths case, the set of costats X of poblem (CHMOILP) ca be ewtte the detemstc fom as: whee X = R a E{b } + K α Va{b }, =,,..., m, 0 ad tege, =,,.., = K α s the stadad omal value such that Φ( K α ) = α ; ad Φ (a) epesets the cumulatve dstbuto fucto of the stadad omal dstbuto evaluated at a. Thus, poblem (CHMOILP) ca be udestood as the followg detemstc veso of a multobectve tege lea pogammg poblem: (MOILP): ma [f (), f (),., f k ()], subect to X. Now t ca be obseved, fom the atue of poblem (MOILP) above, that a sutable scalazato techque fo teatg such poblems s to use the - costat method []. Fo ths pupose, we cosde the followg tege lea pogammg poblem wth a sgle-obectve fucto as: P s (ε): ma f s (), subect to X( ε ) = { R f () ε, K { s }, X' }

4 whee s K={,,,k} whch ca be take abtay. It should be stated hee that a effcet soluto fo poblem (CHMOILP) ca be foud by solvg the scala poblem P s (ε) ad ths ca be doe whe the mmum allowable levels (ε, ε,, ε s-, ε s+,, ε k ) fo the (k-) obectves (f, f,, f s-, f s+,, f k ) ae detemed the feasble ego of solutos X(ε). It s clea fom [] that a systematc vaato of ε 's wll yeld a set of effcet solutos. O the othe had, the esultg scala poblem P s (ε) ca be solved easly at a ceta paamete ε= ε usg the bach-ad boud method [4]. If X(ε) s a uque optmal tege soluto of poblem P s (ε), the becomes a effcet soluto to poblem (CHMOILP) wth a pobablty level α, ( =,, m).. A PARAMETRIC STUDY ON PROBLEM (CHMOILP) Now ad befoe we go ay futhe, we ca ewte poblem P s (ε) the followg scala elaed subpoblem whch may occu the bach-ad-boud pocess as: P s ' (ε): ma f s (), subect to X s (ε), whee R X ( ε) = f () ε, K {s}, g () = a C, =,,...,m, γ s, = β whee the costat β, J {,,.. }, J {,,.. } γ s a addtoal costat o the decso vaable ad that has bee added to the set of costats of poblem P s (ε) fo obtag ts optmal tege soluto by the bach-ad-boud algothm. I addto, t s supposed that: C E{b } + K Va{b }, ( =,,...m). = α I what follows, deftos of some basc stablty otos ae gve fo the elaed poblem P ' s (ε) above. We shall be essetally coceed wth thee basc otos: the set of feasble 4

5 paametes; the solvablty set ad the stablty set of the fst kd (SSK). The qualtatve ad quattatve aalyss of these otos have bee toduced detals by Osma [6, 7] fo dffeet classes of paametc optmzato poblems. Moeove, stablty esults fo such poblems have bee deved. The feasblty codto fo poblem P ' s (ε) s gve the followg. The Set of Feasble Paametes Defto. The set of feasble paametes of poblem P s ' (ε), whch s deoted by A, s defed by: k A = { ε R X ( ε) Φ}. s The Solvablty Set Defto. The solvablty set of poblem P s ' (ε), whch s deoted by B, s defed by: B = { ε A P oblem Ps ( ε) has optmal tege soluto}. The Stablty Set of the Fst Kd Defto 4. Suppose that ε B wth a coespodg optmal tege soluto, the the stablty set of the fst kd of poblem P s ' (ε) coespodg to, whch s deoted by S(), s defed by: ' S() = { ε B ema optmal tege soluto of poblem Ps ( ε) }.. Utlzato of the Kuh-Tucke Necessay Optmalty Codtos fo P ' s (ε). Now, gve a optmal pot, whch may be foud as descbed eale Secto, the questo s: Fo what values of the vecto ε the Kuh-Tucke ecessay optmalty codtos fo the subpoblem P ' s (ε) ae satsfed? I the followg, the Kuh-Tucke ecessay optmalty codtos coespodg to poblem P s ' (ε) wll have the fom: 5

6 k m f s () f s () g () + µ δ + u v (=,,...,) = = s f () ε, K {s}, g () C, ( =,,...m), β, I {,,...,}, γ, J {,,...,}, µ [ f () + ε ] K {s}, δ[g () C ] ( =,,...m), () u [ +β ] I {,,...,}, v [ γ ] J {,,...,}, µ 0, K {s}, δ 0, ( =,,...m), u 0, I {,,...,}, v 0, I {,,...,}, whee I J {,, }, I J = Φ ad all the above elatos of system () above ae evaluated at the optmal tege soluto. The vaables µ, δ, u, v ae the lagaga multples. The fst ad last fou elatos of the system () above epeset a Polytope µδ u v space fo whch ts vetces ca be detemed usg ay algothm based upo the smple method, fo eample, Balsk []. Accodg to whethe ay of the vaables µ, K-{s}, δ, (=,, m), u,( I) ad v, ( J) s zeo o postve, the the set of paametes ε's fo whch the Kuh-Tucke ecessay optmalty codtos ae utlzed wll be detemed. Ths set s deoted by T(). Detemato of the Set T() I what follows, we popose a algothm sees of steps to fd the set of possble ε whch wll be deoted by T(). Fo the set T(), the pot emas effcet fo all values of the vecto ε. Clealy, T() S() The suggested algothm ca be summazed the followg mae. The Algothm: Step. Deteme the meas E{b } ad Va{b } ( =,, m). Step. Covet the ogal set of costats X of poblem (CHMOILP) to the equvalet set of costats X. Step. Fomulate the detemstc multobectve tege lea poblem (MOILP) coespodg to poblem (CHMOILP). 6

7 Step 4. Fomulate the tege lea poblem wth a sgle-obectve fucto P s (ε). Step 5. Solve k-dvdual tege lea poblem P, ( =,,,k) whee P : ma f (), (=,,,k), subect to X, to fd the optmal tege solutos of the k-obectves. Step 6. Costuct the payoff table ad deteme, M (the smallest ad the lagest umbes the th colum the payoff table). Step 7. Deteme the ε 's fom the fomula: t ε = + (M ), K {s} N whee t s the umbe of all pattos of the teval [, M ]. Step 8. k Fd the set I = { ε R ε M, K {s} } Step 9. Choose ε I ad solve the tege lea poblem P s (ε) usg the bach-ad-boud method [4] to fd ts optmal tege soluto. Step 0. Deteme the set T() by utlzg the Kuh-Tucke ecessay optmalty codtos () coespodg to poblem P s ' (ε). Step. If T () s a oe-pot set, go to step. Othewse, go to step. k Step. Defe T ( ) = { ε R ε ε M, K { s} } pespecfed postve eal umbe., whee s ay small Step. Deteme I T (). If I T () = φ, stop. Othewse, go to step 4. Step 4. Choose aothe ε = ε I T () ad go to step 9. The above algothm temates whe the age of I s fully ehausted. 4. AN ILLUSTRATIVE EXAMPLE Hee, we povde a umecal eample to clafy the developed theoy ad the poposed algothm. The poblem ude cosdeato s the followg bcteo tege lea pogammg poblem volvg adom paametes the ght-had sde of the costats (CHBILP). (CHBILP): ma F() = [f (), f ()], 7

8 subect to P{ + b } 0.90, P{- + b } 0.95, P{ + b } 0.90,, 0 ad teges. whee f () = +, f () = +. Suppose that b, ( =,, ) ae omally dstbuted adom paametes wth the followg meas ad vaaces. E {b } =, E{b } =, E{b } = 9, Va {b } = 5, Va {b } = 4, Va {b } = 4, Fom stadad omal tables, we have: K α = K α = K , K α = K Fo the fst costat, the equvalet detemstc costat s gve by: Fo the secod costat: Fo the thd costat: + C = E{b } + K α Va{b } = +.85(5) = C = E{b } + Kα Va{b } = +.645() = C = E {b } + Kα Va{b } = 9+.85() =.57 Theefoe, poblem (CHBILP) ca be udestood as the coespodg detemstc bcteo tege lea pogammg poblem the fom: (BILP): ma [f () = +, f () = + ], subect to , , +.57, 0 ad teges. Usg the ε-costat method [], the poblem (BILP) above wth a sgle-obectve fucto becomes: P (ε): ma f () = +, subect to 8

9 + ε, , , +.57, 0 ad teges. It ca be show easly that.7775 ε Poblem P (ε) ca be solved at ε = ε = usg the bach-ad-boud method [4] ad ts optmal tege soluto s foud (, )= (, 6). Futhemoe, poblem P (ε) ca be ewtte the followg paametc fom as: P '(ε): ma f () = +, subect to + ε, , , , 0 6 Theefoe, the Kuh-Tucke ecessay optmalty codtos coespodg to poblem P '(ε) wll take the followg fom: + µ δ + µ δ δ δ ( ( + δ + δ µ ( δ ( µ, δ +, δ δ + δ u u ), δ ).57) ( ( u u 7.45, ) 6), u, u, 6.9, ε.57,, 6, + ε ) 0 (#) 9

10 whee all the above epessos of system (#) ae evaluated at the optmal tege soluto (, ) = (, 6). I addto, t ca be show that δ = δ = δ u, u > 0, µ 0 Theefoe, the set T (, 6) s gve by: T (, 6) = {ε R.7775 ε }. A systematc vaato of ε R ad.775 ε wll yeld aothe stablty set T (, 6), ad so o. 5. CONCLUSIONS The geeal pupose of ths study was to vestgate stablty of the effcet soluto fo chace-costaed multobectve tege lea pogammg poblem. A paametc study has bee caed out o the poblem ude cosdeato, whee some basc stablty otos have bee defed ad chaactezed fo the fomulated poblem. May aspects ad geeal questos ema to be studed ad eploed the feld of multobectve tege optmzato poblems ude adomess. Ths pape s a attempt to establsh udelyg esults whch hopefully wll help othes to aswe some o all of these questos. Thee ae howeve seveal usolved poblems, ou opo, to be studed futue. Some of these poblems ae: () A algothm s equed fo solvg multobectve tege lea pogammg poblems volvg adom paametes the left-had sde of the costats, () A algothm s eeded fo teatg lage-scale multobectve tege lea olea pogammg poblems ude adomess, () A algothm should be hadled fo solvg tege lea ad tege olea goal pogams volvg adom paametes. Refeeces: [] Balsk, M., (96), A Algothm Fo Fdg All Vetces of Cove Polyhedal Sets, SIAM Joual, Vol. 9, No., [] Chakog, V. ad Hames, Y. Y., (98), Multobectve Decso-Makg: Theoy ad Methodology (Noth Hollad Sees System Scece ad Egeeg). [] El-Baa, A. Z. ad Youess, E. A., (99), O Some Basc Notos of Stochastc Multobectve Poblems wth Radom Paametes the Costats, Mcoelectocs. Relablty, Vol., No.,

11 [4] Guddat. J., Vasquez, F., Tamme, K. ad Wedle, K., (985), Multobectve ad Stochastc Optmzato Based o Paametc Optmzato, Akademe-Velage, Bel. [5] Osama, Ez-Eld., (000), O Stochastc Multobectve Itege Lea Pogammg Poblems, M.SC. Thess, Helwa Uvesty, Cao, Egypt. [6] Osma, M. S. A., (977), Qualtatve Aalyss of Basc Notos Paametc Cove Pogammg I (Paametes the Costats), Appled. Math. CSSR Akad. Ved. Pague,, 8-. [7] Osma, M. S. A., (977), Qualtatve Aalyss of Basc Notos Paametc Cove Pogammg II (Paametes the Obectve Fucto), Appled. Math. CSSR Akad. Ved. Pague,, -48. [8] Osma, M. S. A. ad Saad, O. M., (994), O the Soluto of Chace-Costaed Multobectve Lea Pogammg Poblems wth A Paametc Study, Poceedgs of the Fst Iteatoal Cofeece O Opeatos Reseach ad ts Applcatos, Hghe Techologcal Isttute, Ramada Teth. Cty, Egypt. [9] Saad, O. M. ad Ktta, H. F., (00), Multobectve tege lea pogammg poblems ude adomess, IAPQR TRANSACTIONS, Vol.8, No., [0] Saad, O. M., (006), Optmzato ude ucetaty: A State-of-the-At, Pape accepted fo publcato Appled Mathematcs ad Computato, to appea. [] Seppälä, Y., (988), O Accuate Lea Appomatos Fo Chace-Costaed Pogammg, Joual of Opeatoal. Reseach. Socety, Vol. 9, No. 7, [] Shaf, W.H. ad Saad, O.M., (005), O stablty multobectve tege lea pogammg: A stochastc appoach, Ameca Joual of Appled Sceces, Vol., No., [] Stochastc Pogammg Web Ste, (00), Stopog.og, Cuet as of July 00. [4] Taha, H. A., (975), Itege Pogammg: Theoy, Applcatos ad Computatos, Academc Pess, New Yok. [5] Vogel, S., (99), O Stablty Multobectve Pogammg-A Stochastc Appoach, Mathematcal Pogammg, 60, 9-9.